Approximations for Performance of Energy Detector and $p$-norm Detector

Although the approximations based on the central limit theorem (CLT) for the detection probability (Pd) of the energy detector and of its more generalized cousin, the p-norm detector, are accurate for large sample sizes, their accuracy is poor otherwise. A recent work has addressed this problem by developing a cube-of-Gaussian approximation (CGA). However, CGA may not be the only option and thus this letter investigates five other classical approximations for deriving Pd. They tightly match the exact values even for few samples and thus are more accurate than CLT. These approximations have been unnoticed in the signal detection research community and this letter aims at making them known to a wider audience. Because the range of their potential applications could be diverse, to demonstrate their utility, we derive a novel expression for the area under the receiver operating characteristic curve of the p-norm detector.